4 Answers
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However, Bitcoin balances are stored in uniquely identifiable "transaction outputs" that can only be spent by the owner of the recipient address.

Example:
Alice sends Bob 1 BTC, Bob uses it to send another payment.

When Alice sends Bob a payment of 1 BTC, she signs a transaction that deducts 1 BTC from her funds and creates a new transaction output that is worth 1 BTC and can only be spent by Bob, the owner of the recipient address.

Bob now wants to send 0.4 BTC to Charles. The transaction output from Alice's transaction is now used to fund this new transaction. The transaction creates two new outputs: One with 0.4 BTC that is associated with Charles' address, and one with 0.6 BTC associated with Bob's address (it is the change). The first transaction output (from Alice's transaction) is consumed by the transaction.

Unspent Transaction Outputs are identified by the (unique) hash of the transaction they were created in, and their position in the list of outputs.

Can each Bitcoin and Satoshi be uniquely identified?

1. Does each Satoshi have it's own unique identity?

No.

If I show you transaction xyz on blockchain.info, you can see where the groups of satoshi's have gone (the transaction outputs) and you can see where the satoshi's have come from (the inputs) BUT you cannot identify which of the input satoshi's are given to each of the outputs.

You can also click on each input to find the transaction whose inputs were spent to create that output, and you can do this all the way back to the coin genesis every time but every time there is an input and an output all of the satoshi's are grouped so you cannot find an individual satoshi.

Each user of bitcoin need have at least one address to receive coins. I just created an address 18TGt8oGgLzkCpjxMbiRfvbDo9NqGisopp, which due to the properties of how it is generated, is presumed to be unique.

If someone sends me any bitcoin they will be taking a coin from an existing balance on THEIR account (another bitcoin address) which will need to be signed using their private key. Once the transaction has been broadcast to the network, it will eventually make it into a block which will allow you to then spend it by signing it over to someone else.

While balances are generally expressed to users in whole bitcoin units, or sub units such as bitmil, transactions are expressed in satoshis. For example, if you were to give me a 1 bitcoin tip for this awesome answer :P the transaction would be of 100000000 to 18TGt8oGgLzkCpjxMbiRfvbDo9NqGisopp.

I'm aware of wallet addresses, but I want to know if the coins themselves are uniquely identifiable. I'm assuming they must be identifiable in one way or another in order to prevent forgeries etc?
– series0neJan 31 '14 at 14:52

When you spend a coin you are spending a previous output (e.g. 100000000 to 18TGt8oGgLzkCpjxMbiRfvbDo9NqGisopp from 1anotherbitcoinaddress or from generate), which becomes the input for the current transaction. Each unique satoshi need not be identified but the input(s) needs be greater or equal to the output(s).
– MarkJan 31 '14 at 14:58

No, coins are digital divisible units even when it's noun refers to a physical indivisible unit.
There are nothing like I have 1/2 of this coin and 1/4 of that other indeed even the balance concept it self is ill defined in bitcoin due to multiple signature transanctions and other type of scripts.

Your balance is just the amount of Bitcoin that you can spend with the private keys you have.

On the other side as every transaction is traceable in the blockchain you can follow the trace of every generation transaction (the one that gives you the block reward) but there is nothing like a coin id.